So, in this optional module, we'll look at the two key player problems again. The goal of the first problem is to identify a small set of nodes, whose removal will create maximum disruption. Now, in this case, a traditional centrality algorithm may not work, because the optimization goal is to break up the network. So we need a quantitive measure of the breakage. If dij is the distance between nodes i and j, then 1 over dij is the closeness of these two nodes. If we add the closeness of all nodes and normalize it by the number of node pairs, we'll get a measure of cohesiveness as a fraction. So, 1 minus this value is a measure of fragmentation. Our goal is to maximize this fragmentation metric. In the model terrorist network shown here, removing the red nodes A, B, and C will break up the network into seven components, with F reaching a value of 0.59. The second key player problem, is trying to find a group of S influencers, which can reach a maximum number of nodes within K steps. The number of unique nodes reachable from a starting node is called the reach, of the starting node. For this we need to adapt the concept of reach to limit it to k steps. We also need to adapt it to measure the distance of an arbitrary node from a group of nodes between our influences. The distance from one node to a group of nodes can be defined as a maximum, or average, or minimum distance, of the node from the members of the group. Often, the minimum distance is a good choice. So the distance, we could reach, can then be through of, as the proportion of all nodes reached by the group, where the nodes are weighted by the distance from the set. And only nodes at distance 1 are given full rate. Hence, a distance we could reach that use a maximum value of 1, where every outside node is adjacent to at least one member of the set of influences. In the network shown, just three nodes, A, C, and D, are sufficient to reach every other member within just four steps. Now this concludes this module, where we looked at several analytic techniques and measures to extract different kinds of insights from a network.